Vacuum stability and scalar masses in the superweak extension of the standard model

We study the allowed parameter space of the scalar sector in the superweak extension of the standard model (SM). The allowed region is defined by the conditions of (i) stability of the vacuum and (ii) perturbativity up to the Planck scale, (iii) the pole mass of the Higgs boson falls into its experimentally measured range. We employ renormalization group equations and quantum corrections at two-loop accuracy. We study the dependence on the Yukawa couplings of the sterile neutrinos at selected values. We also check the exclusion limit set by the precise measurement of the mass of the W boson. Our method for constraining the parameter space using two-loop predictions can also be applied to simpler models such as the singlet scalar extension of the SM in a straightforward way.


I. INTRODUCTION
Currently particle physics is in a similar situation as physics was about 120 years ago. Its standard model (SM) can explain successfully most of the low and high energy phenomena and provide predictions that are in agreement with measurements at high precision. Nevertheless, there are also a handful of outstanding observations that cannot be predicted by the standard model and point towards beyond the standard model (BSM) physics. These unexplained facts are (i) the non-vanishing neutrino masses and mixing matrix elements [1,2], (ii) the metastable vacuum of the standard model [3,4], (iii) the need for leptoand/or baryogenesis to explain baryon asymmetry, i.e. our obvious existence, (iv) the existence of dark matter in the Universe [5][6][7][8][9], and also (v) the existence of dark energy in the Universe [5]. In addition there is general consensus about the occurrence of cosmic inflation in the early Universe, which also calls for an explanation. There are other observations in particle physics that have almost reached the status of discoveries. Most prominently the prediction of the standard model for the anomalous magnetic moment a µ of the muon [10] is smaller than the result of the measurement [11,12] by 4.2 standard deviations. In this case however, the status of the theory is controversial because the evaluation of the hadronic contribution to a µ requires non-perturbative approach, and the result depends on the method [10,13]. The resolution of this discrepancy calls for an independent evaluation of this hadronic vacuum polarization contribution before discovery can be claimed. Some of the observations (i-v) should find understanding in particle physics models, while others may have cosmological origins. Nevertheless, the intimate relation between particle physics and the early Universe, originating from the universal expansion of spacetime, gives a strong support for searching answers within particle physics by extending the SM. Such extensions can be put into three categories: (a) ultraviolet complete models from theoretical motivations, such as supersymmetric models; (b) effective field theories like the standard model effective field theory (SMEFT); (c) simplified models that focus on a subset of open questions. This third category includes the dark photon models (gauge extension, see e.g. Refs. [14,15]), the singlet scalar extensions (see e.g. Refs. [16][17][18]) and the introduction of neutrino mass matrices with some variant of the see-saw mechanism, such as in Ref. [19].
The UV complete supersymmetric extensions of the SM are very attractive for solving theoretical issues, but they are becoming less favored by the results of the LHC experiments [20,21]. Effective field theories proved to be very useful in the past. However, the SMEFT contains 2499 dimension six operators [22], which makes it rather difficult to study experimentally. The simplified models on the other end contain only few new parameters, hence are very attractive from the experimental point of view. However, being simplified models, those cannot give answers to all observations pointing towards BSM physics simultaneously.
In this paper we study a simple UV complete BSM extension along the principles of the SM itself: a renormalizable gauge theory that adds one layer of interactions below the hierarchic layers of the strong, electromagnetic and weak forces, which is called superweak (SW) force [23], mediated by a new U(1) gauge boson Z , see Fig. 1. In order to explain the origin of neutrino masses, the field content is enhanced by three generations of right-handed neutrinos. The new gauge symmetry is broken spontaneously by the vacuum expectation value of a new complex scalar singlet. According to exploratory studies, the superweak extension of the standard model (SWSM) has the potential to explain the origin of (i) neutrino masses and mixing matrix elements [24], (ii) dark matter [25], (iii) cosmic inflation [26], (iv) stabilization of the electroweak vacuum [26] and possibly (v) leptogenesis (under investigation).
While these findings are promising, more refined analyses are needed in order to explore the viability of the model. The main motivation of our work is not to prove that the SWSM is listed above can be answered within a single model with as few new parameters as possible.
In this paper we revisit the study of the parameter space of the scalar sector of the SWSM as allowed by the requirement of the stability of the vacuum. We improve significantly on our previous analysis [26] in two respects. Firstly, we use renormalization group equations (RGEs) containing the beta functions at two-loop order. More importantly, we take into account both the radiative corrections up to two-loop accuracy and the measured physical values and uncertainties of the parameters of the scalar sector as constraints. A similar study has been performed earlier in the simplified model of single real scalar extension of the SM in Ref. [18]. The important difference between the present work and that analysis is that we include the effect of the right-handed neutrinos in the running of the couplings, which constrains the parameter space further. The inclusion of the two-loop effects is also for the first time in the present work.

II. SUPERWEAK MODEL
The SWSM is a gauged U(1) extension of the standard model with an additional complex scalar field χ and three families of sterile neutrinos ν R,i . The model was defined in Ref. [23] and further details on the new sectors were presented in Refs. [24,26]. Here we recall some details relevant to the present analysis.
The anomaly free charge assignment is shown in Table I. In particular, the χ field does not couple directly to any fields of the SM.
After spontaneous symmetry breaking (SSB), we parametrize the SM scalar doublet φ and the new scalar field as where v and w are the two vacuum expectation values (VEVs), H and S are two real, scalar fields and σ + , σ φ/χ are charged and neutral Goldstone bosons. In terms of these fields the scalar potential in the SWSM is given by The constant V 0 is irrelevant in our considerations, so we set it to zero in the rest of the paper. Substituting the parametrization (1) into (2), we obtain the tree-level (effective) of the real scalar fields. The VEVs are determined by the tadpole equations: The mass matrix of the scalar fields is given by the Hessian: which can be diagonalized by a rotation matrix among the scalar couplings and VEVs. Explicitly, the angle of rotation and the scalar masses M h and M s can be expressed through the VEVs and couplings at tree level as In the absence of mixing (λ = 0, θ s = 0) we have M h = 2λ φ v 2 , M s = 2λ χ w 2 . As the scalar fields are coupled to the W ± bosons with the interaction vertices Γ µν hW W = i 2 g 2 L v cos θ s g µν , and Γ µν sW W = only the BEH field is coupled to the W bosons and to the other SM fields in the limit of vanishing mixing between the scalars. Hence, we naturally identify the VEV v as that related to the Fermi coupling and also the parameter M h with the mass of the Higgs boson measured at the LHC [28] by introducing the notation m h = 125.10 GeV, ∆m h = 0.14 GeV and v = and In accordance with this assumption, we restrict θ s to fall in the range (−π/4, π/4).
The VEV w can be expressed through these known parameters and the scalar couplings using Eqs. (8) and (9), Thus, the formal conditions for the non-vanishing w, required at the electroweak scale are either (the second condition deriving from the positivity constraint in (7) for positive v 2 w 2 ), or As we have fixed v and M h experimentally, the input value of λ φ decides which of these two cases are to be considered.
Eqs. (3)-(5) are valid at tree level. The effect of the quantum corrections can be summarized by substituting the potential V with the effective potential V eff , whose formal loop expansion is.
where V

III. VACUUM STABILITY IN THE SWSM AT ONE-LOOP ACCURACY
The potential (3) is stable if it is bounded from below. Due to its continuity in the field variables, it is sufficient to study the positivity of (3) for large values of h and s, which translates to the following conditions on the quartic scalar couplings: Taking into account the radiative corrections leads to (i) dependence on the renormalization scale µ for all renormalized couplings and (ii) the corrections V (i) eff . While it is straightforward to require that the conditions (17) be satisfied for the running couplings at any sensible value of µ, we cannot write the stability conditions for the one-loop effective potential in a closed form such as in Eq. (17) valid at tree level. Instead, we take an alternative path by requiring the existence of a non-vanishing w(M t ) indirectly, extracting it from the known pole mass of the Higgs boson, rather than computing it explicitly form the effective potential (16) with radiative corrections taken into account. Our procedure can be described in terms of analytic expressions at the one-loop accuracy as follows.
We investigate the vacuum stability in the range µ ∈ (M t , M Pl ), i.e. from the pole mass M t of the t quark up to the Planck mass M Pl where quantum gravitational effects become important. The scale dependence of a given coupling g is described by the autonomous system coupled differential equations of the form called RGEs, where ∂/∂t = µ ∂/∂µ. We assume that the model remains perturbatively valid for the complete range by requiring for any coupling g in the theory, which we check in the stability analysis. Consequently, we can employ perturbation theory to compute the β g functions. We integrate the complete set of RGEs of the SWSM, while requiring the stability and perturbativity conditions (17) and (19). We also assume the existence of w at the scale µ = M t , which implies the existence of a second massive neutral gauge boson and a second massive scalar particle as predictions of the model. To check this condition, we compute the loop corrected scalar mixing angle and scalar pole masses: using the shorthand notation for w   21) and (22)).
The complete set of running couplings can be grouped into three sets. The (i) SM couplings g Y , g L , g s , y t , the (ii) SW gauge coupling g z and (iii) the scalar quartic couplings λ φ , λ χ , λ together with the sterile neutrino Yukawa coupling y x . We assume one light sterile neutrino -a candidate for dark matter [25] -and two heavy ones with equal masses for simplicity, y x = y x,5 = y x,6 . We neglect the effect of the SW gauge coupling from our analysis because its maximally allowed value is very small, g z 10 −4 , if the model is to explain the origin of dark matter [25] and also should obey the direct observational limit of the NA61 experiment [29]. Explicitly, in group (iii) we have the following autonomous system of RGEs at one loop: for the scalar couplings, with β s , using w as a free input parameter. Starting from the initial value w = w (1) (M t ), we searched for the w at which which we call w = w (2) . This procedure of starting with using only such points in the parameter space where the condition w  Also, the volume V λ (y x ) increases slightly with increasing order in perturbation theory.
We have checked Eq. (24) numerically for both cases (a) and (b) at randomly selected input values {λ φ , λ χ , λ, y x } µ=Mt in the range µ ∈ 0.5M t , 2M t and compared the scale dependences of the tree level masses (9) and (10) to the scale dependences of the one-loop accurate pole masses (21) and (22). As shown in Fig. 7, we have found that the scale dependences of the tree level masses are reduced significantly at one-loop and even more at two-loop accuracy. The sizeable difference between the scalar pole masses M s at the first two orders of perturbation theory (and to much less extent between the next two orders) are not caused by radiative corrections. Rather than loop corrections to the masses, the jumps originate from the shifts in w(M t ) required to reproduce Higgs boson pole mass at different orders of perturbation theory, as can be seen in Fig. 2.
The theoretical prediction for the W -boson mass uses precision electroweak observables (except M W itself) and it is sensitive to new physics [34]. Hence, it is often used as a We computed the SW contributions δM SW W to M theo.
W at one loop accuracy. We found, that the contribution of the new gauge sector is heavily suppressed due to the required smallness of the new gauge coupling g z 10 −4 . The sterile neutrinos may change the measured value of the Fermi coupling G F affecting the mass of the W boson already at tree level [38]. As a matter of fact right-handed neutrinos can provide significant contribution to the W boson mass [39], although at the price of introducing some tension with universality bounds. Hence, a proper account of the effect of sterile neutrinos is certainly warranted, but it is beyond the scope of the present paper and we leave it for a planned global scan of the parameter space.
The contribution of the new scalar sector to M W however, can be comparable to ∆M W [34,40]. We present the SW correction δM W in ( where the new scalar particle is allowed by vacuum stability, perturbativity and precision measurement of the Higgs boson mass. We see that the W -mass measurement at present uncertainties does not provide significant reduction of the parameter space. However, if the improved measurement published recently by the CDF collaboration [41] will be confirmed, the stability of the vacuum in the high-mass region becomes incompatible with the CDF-II W mass as in that case the SW correction to the SM value is negative. The low-mass region also becomes significantly constrained. In Fig. 9 we present the allowed parameter space together with contour lines representing the border of the excluded parameter space (below the line) assuming a δM W increase of the W mass by selected benchmark values due to the new scalar in the self energy loop. Clearly, the large positive shift required to explain the CDF-II result is not compatible with the conditions of stability and perturbativity of the scalar sector of the model.

V. CONCLUSIONS AND OUTLOOK
In this paper we have scanned the parameter space of the superweak extension of the standard model in order to find the allowed parameter space of the scalar sector where the following assumptions are fulfilled: (i) the vacuum be stable and (ii) the model parameters remain perturbative up to the Planck scale, (iii) the pole mass of the Higgs boson must fall into its experimentally measured range. The first two of these constraints were taken into account in our preliminary work [26]. In this paper we superseed that former study by Of course, there is a lot of experimental results that also constrain the parameter space.
The new physics contributions to electroweak precision observables as well as direct searches for the decay of a scalar particle into standard model ones provide strong constraints. Of those, we have studied only the effect of the experimental result on the mass of the W boson in this paper. We saw that while M W can indeed limit the parameter space, the current world average without the new CDF-II result cannot provide further constraint on the parameter space. If we also include the CDF II result in the average -in spite of being incompatible with the previous average -, then the parameter space allowed by our assumptions become incompatible with the W -mass constraint. Clearly, it is of utmost importance to take into account all the available experimental constraints, not only from collider experiments, also from neutrino experiments. Such a complete study of the parameter space is beyond the scope of the present paper and we leave it to an upcoming study where we plan to use the analytic expressions of the present work.

ACKNOWLEDGMENTS
We are grateful to members of the ELTE PPPhenogroup (pppheno.elte.hu/people), especially to Josu Hernandez-Garcia for useful discussions. This work was supported by grant K 125105 of the National Research, Development and Innovation Fund in Hungary.
Appendix A: Loop corrections to the scalar masses in the SWSM On one hand, the quantum effects correct the tadpole equations (4), such that where T ϕ is the sum of all 1PI one-point functions with external leg ϕ = H or S. On the other hand, the scalar self-energies Π ϕ I ϕ J -i.e. the sum of all 1PI Feynmann graphs with external legs ϕ I and ϕ J -correct directly the propagator matrix of the scalar fields H and S. The inverse-propagator matrix after applying the tadpole equations (A1) to eliminate the mass parameters µ φ and µ χ is given as whereΠ ϕϕ (p 2 ) = Π ϕϕ (p 2 ) − T ϕ / ϕ with ϕ = v, w. In order to obtain the mixing angle θ s (20) and the scalar pole masses (21) and (22), one has to diagonalize the real part of (A2), leading to Eqs. (20), (21) and (22).
We have implemented a SARAH model file for the SWSM [24], and used it to compute the one-loop scalar self-energies and tadpoles in Feynman gauge (ξ = 1). In the following, we list explicitly the one-loop corrections to the scalar inverse-propagator matrix (A2), after some simplifications of the SARAH output.
The Higgs self energy is where κ = (4π) −2 and m p (µ) is the running mass of particle p as computed using tree level and In the special case of vanishing masses, the latter reduces to where Θ is the Heaviside step function. In this work, we are not concerned with the decay width Γ h/s of the scalar bosons. The imaginary parts of the one-loop self energies and tadpoles contribute to the decay widths, and we neglect those completely. The SWSM introduces new corrections to the W and Z gauge boson self-energies. The radiative corrections from the new gauge sector are neglected due to coupling suppression g z 10 −4 , whereas the sterile neutrinos may not only contribute radiatively through the PMNS matrix but also contribute at tree level by affecting the Fermi coupling G F through the low energy muon decay. We neglect the neutrino contributions (to be investigated in an upcoming paper) and focus here on the pure scalar radiative corrections. In the MS scheme the scalar SW contribution to the gauge boson self-energies is where the loop function F is defined as (B2) We have checked in the R ξ gauge, that the scalar contribution Π SW V V (p 2 ) is explicitly independent of the gauge parameter ξ, hence gauge invariant. Furthermore, Π SW V V (M 2 V ) is independent of the the renormalization scale at one loop accuracy.
The shift in the electroweak input parameters due to the SW corrections is then which agrees with Eq. (22) of [18]. The W boson pole mass is then given as